Hybrid Godunov-Glimm Method for a Nonconservative Hyperbolic System with Kinetic Relations

We study the numerical approximation of a system from the physics of compressible turbulent flows, in the regime of large Reynolds numbers. The PDE model takes the form of a nonconservative hyperbolic system with singular viscous perturbations. Weak solutions of the limit system are regularization dependent and classical approximate Riemann solvers are known to grossly fail in the capture of shock solutions. Here, the notion of kinetic functions is used to derive a complete set of generalized jump conditions which keeps a precise memory of the underlying viscous mechanism. To enforce for validity these jump conditions, we propose a hybrid Godunov-Glimm method coupled with a local nonlinear correction procedure.